Generalized Mixed Models for Ordinal Longitudinal Outcomes using PROC GLIMMIX

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Generalized Mixed Models for Ordinal Longitudinal Outcomes using PROC GLIMMIX SAS Data Manipulation: * Reading in all data; DATA alldata; SET annk.annknewfinal; WHERE NMISS(age80, mmse16)=0; cam012=cam; LABEL cam012= " Some, Full Delirium"; run; * Observed frequencies; PROC FREQ DATA= alldata; TABLE cam012; run; Some, Full Delirium Cumulative Cumulative cam012 Frequency Percent Frequency Percent ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ none 0 190 37.25 190 37.25 some 1 239 46.86 429 84.12 full 2 81 15.88 510 100.00 1a) Ordinal Single-Level Empty Means Model for None vs. Some vs. Full Delirium via GLIMMIX (Composite Model Form) Note: it appears ILINK does not work correctly for CLOGIT models. Logit y 0 Logit y 1 ti 001 ti 002 TITLE1 "GLIMMIX Ordinal Single-Level Empty Means Model"; TITLE2 "Cumulative Logit Link, Multinomial Distribution"; PROC GLIMMIX DATA=alldata METHOD=QUAD; MODEL cam012 (DESCENDING) = / SOLUTION LINK=CLOGIT DIST=MULT; run; Response Profile Ordered Total Value cam012 Frequency 1 2 81 2 1 239 3 0 190 The GLIMMIX procedure is modeling the probabilities of levels of cam012 having lower Ordered Values in the Response Profile table. This means we are predicting up, not down. Convergence criterion (ABSGCONV=0.00001) satisfied. -2 Log Likelihood 1035.58 AIC (smaller is better) 1039.58 AICC (smaller is better) 1039.60 BIC (smaller is better) 1048.05 CAIC (smaller is better) 1050.05 HQIC (smaller is better) 1042.90 Parameter Estimates Intercept 2-1.6670 0.1211 508-13.76 <.0001 1.22E-14 Intercept 1 0.5213 0.09159 508 5.69 <.0001-71E-15 Intercept2 = logit of probability of y > 1 = exp( 1.667) / (1+exp( 1.667)) =.1588 Intercept1 = logit of probability of y > 0 = exp( 0.521) / (1+exp( 0.521)) =.6274 (=.4686+.1588) Probability(y=0) = 1 intercept1 = 1.6274 =.3725 Probability(y=1) = intercept1 intercept2 =.6274.1588 =.4686 Probability(y=2) = intercept2 0 =.1588 0 =.1588 Ordinal Longitudinal Example 1 of 7

1b) Ordinal Empty Means, Random Intercept Only Model (same random intercept both sub-models) Logit yti 0 001 U0i Logit y 1 U ti 002 0i TITLE1 "GLIMMIX Ordinal Empty Means, Random Intercept Model"; TITLE2 "Cumulative Logit Link, Multinomial Response Distribution"; MODEL cam012 (DESCENDING) = / SOLUTION LINK=CLOGIT DIST=MULT; RANDOM INTERCEPT / TYPE=UN SUBJECT=patient_ID; COVTEST "Need Random Intercept?" 0; * Test if G matrix (1,1)=0; RUN; run; -2 Log Likelihood 978.26 AIC (smaller is better) 984.26 AICC (smaller is better) 984.31 BIC (smaller is better) 991.79 CAIC (smaller is better) 994.79 HQIC (smaller is better) 987.30 ariance Parameter Estimates Parm Subject Estimate Error Gradient UN(1,1) PATIENT_ID 1.2547 0.3480-0.00002 Solutions for Fixed Effects Intercept 2-2.0423 0.1920 90-10.63 <.0001-0.00008 Intercept 1 0.6855 0.1640 90 4.18 <.0001 0.000062 Tests of ariance Parameters Based on the Likelihood Label DF -2 Log Like ChiSq Pr > ChiSq Note Need Random Intercept? 1 1035.58 57.32 <.0001 MI This matches the 2ΔLL MI: P-value based on a mixture of chi-squares. The fixed effects for the intercepts are not the same as in the previous model. This is for two reasons: (1) They are conditional on the random intercept (i.e., expected proportions for someone with U 0i = 0). So for that kind of person, the probability of y > 1 = exp( 2.0411) / (1+exp( 2.0411)) =.115 (not.159), and the probability of y > 0 = exp( 0.6855) / (1+exp( 0.06855)) =.665 (not.627). (2) They are scaled differently because there is now more total variance in the outcome: Total variance = Var(U 0i ) + Var(e ti ) = 12547 + π 2 /3 = 1.2547 + 3.29 = 4.539 = only 3.29 in previous single-level model Ordinal Longitudinal Example 2 of 7

1c) Ordinal Mixed Model: Adding a Fixed Slope and a Random Slope for Days Since Hospital Admission (0=Day Hospitalized) Logit y 0 U Day U Logit y 1 U Day U ti 001 10 1i ti 0i ti 002 10 1i ti 0i TITLE1 "GLIMMIX Ordinal Mixed Model -- Adding a Fixed and Random Slope for Day"; TITLE2 "Cumulative Logit Link, Multinomial Response Distribution"; MODEL cam012 (DESCENDING) = day / SOLUTION LINK=CLOGIT DIST=MULT; RANDOM INTERCEPT day / TYPE=UN SUBJECT=patient_ID; COVTEST "Need Random Slope?". 0 0; * Leave (1,1), test if (2,1) and (2,2) =0; run; -2 Log Likelihood 966.27 AIC (smaller is better) 978.27 AICC (smaller is better) 978.43 BIC (smaller is better) 993.33 CAIC (smaller is better) 999.33 HQIC (smaller is better) 984.34 Fit statistics from Fixed Linear Day, Random Intercept Model: -2 Log Likelihood 973.70 AIC (smaller is better) 981.70 AICC (smaller is better) 981.78 BIC (smaller is better) 991.75 CAIC (smaller is better) 995.75 HQIC (smaller is better) 985.75 ariance Parameter Estimates Parm Subject Estimate Error Gradient UN(1,1) PATIENT_ID 1.9373 0.7856-0.00005 UN(2,1) PATIENT_ID -0.1871 0.1484-0.0004 UN(2,2) PATIENT_ID 0.06592 0.04296-0.00092 Solutions for Fixed Effects Intercept 2-1.8576 0.2582 90-7.20 <.0001-0.0002 Intercept 1 1.0682 0.2437 90 4.38 <.0001 0.000104 day -0.09879 0.05501 84-1.80 0.0761-0.00088 Tests of ariance Parameters Based on the Likelihood Label DF -2 Log Like ChiSq Pr > ChiSq Note Need Random Slope? 2 973.70 7.44 0.0153 MI This matches the 2ΔLL MI: P-value based on a mixture of chi-squares. Interpret the fixed intercept 2: Interpret the fixed intercept 1: Interpret the fixed linear effect of day: Ordinal Longitudinal Example 3 of 7

2a) Ordinal Mixed Model: Effects of MMSE on Intercept and Slope Logit y 0 U Day MMSE 16 MMSE 16Day U Logit y 1 U Day MMSE 16 MMSE 16 Day U ti 001 10 1i ti 01 i 11 i ti 0i ti 002 10 1i ti 01 i 11 i ti 0i Hoffman Psyc 945 Example 6b TITLE1 "GLIMMIX Ordinal Mixed Model -- Effects of MMSE on Intercept and Slope"; TITLE2 "Cumulative Logit Link, Multinomial Response Distribution"; MODEL cam012 (DESCENDING) = day mmse16 day*mmse16 / SOLUTION LINK=CLOGIT DIST=MULT; RANDOM INTERCEPT day / TYPE=UN SUBJECT=patient_ID; COVTEST "Still Need Random Intercept?" 0 0 0; COVTEST "Still Need Random Slope?". 0 0; run; NOTE: GCONV convergence criterion satisfied. FROM THE LOG: NOTE: At least one element of the (projected) gradient is greater than 1e-3. -2 Log Likelihood 941.85 AIC (smaller is better) 957.85 AICC (smaller is better) 958.14 BIC (smaller is better) 977.94 CAIC (smaller is better) 985.94 HQIC (smaller is better) 965.96 ariance Parameter Estimates Parm Subject Estimate Error Gradient UN(1,1) PATIENT_ID 1.5528 0.6970-0.0012 UN(2,1) PATIENT_ID -0.1970 0.1418-0.00588 UN(2,2) PATIENT_ID 0.06289 0.04045-0.00421 Solutions for Fixed Effects Intercept 2-2.0024 0.2614 89-7.66 <.0001-0.00085 Intercept 1 0.9388 0.2427 89 3.87 0.0002 0.00126 day -0.1255 0.05860 83-2.14 0.0352 0.007006 mmse16-0.08364 0.03233 333-2.59 0.0101-0.01616 day*mmse16-0.00822 0.008001 333-1.03 0.3050-0.05209 Tests of ariance Parameters Based on the Likelihood Label DF -2 Log Like ChiSq Pr > ChiSq Note Still Need Random Intercept? 3 986.88 45.03 <.0001 -- Still Need Random Slope? 2 949.46 7.61 0.0141 MI MI: P-value based on a mixture of chi-squares. --: test with unadjusted p-values. Interpret the fixed main effect of MMSE: Interpret the fixed interaction of MMSE by Day: This model makes what is referred as the proportional odds assumption. This means that although we have estimated separate intercepts for the two sub-models, we have constrained the fixed slopes for the predictor effects to be the same for each submodel (as well as the random effects). One alternative is the nominal model, which estimates separate intercepts, fixed effects, and random effects variances for each sub-model (specified as a baseline category vs. each of the other alternatives). Ordinal Longitudinal Example 4 of 7

2b) Attempting the Nominal Model: Sub-models Prob(y=1 if 0 or 1, y=2 if 0 or 2) Logit y 0 U Day MMSE 16 MMSE 16Day U Logit y 1 U Day MMSE 16 MMSE 16 Day U ti 001 101 1i1 ti 011 i 111 i ti 0i1 ti 002 102 1i2 ti 012 i 112 i ti 0i2 TITLE1 "GLIMMIX Nominal Mixed Model -- Effects of MMSE on Intercept and Slope"; TITLE2 "Baseline Logit Link, Multinomial Response Distribution"; CLASS patient_id cam012; MODEL cam012 (REF=FIRST) = day mmse16 day*mmse16 / SOLUTION LINK=GLOGIT DIST=MULT; RANDOM INTERCEPT day / TYPE=UN SUBJECT=patient_ID GROUP=cam012; run; The initial estimates did not yield a valid objective function. ariance Parameter Estimates Parm Subject Group Estimate Error Gradient UN(1,1) PATIENT_ID cam012 1 1.8471.. UN(2,1) PATIENT_ID cam012 1-0.3055.. UN(2,2) PATIENT_ID cam012 1 0.05900.. UN(1,1) PATIENT_ID cam012 2 3.9296.. UN(2,1) PATIENT_ID cam012 2-0.4456.. UN(2,2) PATIENT_ID cam012 2 0.07355.. We have another alternative, in which the fixed effects of predictors are allowed to vary across submodels, but the random effects variances do not. 2c) Ordinal Mixed Model: NON-PROPORTIONAL Effects of MMSE on Intercept and Slope Logit yti 0 001 101 U1i Dayti 011 MMSEi 16 111 MMSEi 16Dayti U0i Logit y 1 U Day MMSE 16 MMSE 16 Day U ti 002 102 1i ti 012 i 112 i ti 0i TITLE1 "Ordinal Mixed Non-Proportional Model for MMSE"; PROC NLMIXED DATA=alldata METHOD=GAUSS TECH=QUANEW GCONV=1e-12; * Must list all parms to be estimated here with start values; * B01 and B02 = intercepts for each equation; * B's = fixed effects, now separate per equation; * V's = variance components in order of G matrix; PARMS B01=.6 B02=-1.6 B11day=0 B21mmse=0 B31mmseday=0 B12day=0 B22mmse=0 B32mmseday=0 V11=1 V21=-.2 V22=.05; * Linear predictive model; Y1 = B01 + B11day*day + U1*day + B21mmse*mmse16 + B31mmseday*mmse16*day + U0; Y2 = B02 + B12day*day + U1*day + B22mmse*mmse16 + B32mmseday*mmse16*day + U0; * Model for probability of response - writing it the shorter way; IF (cam=0) THEN P = 1 - (1/(1 + EXP(-Y1))); ELSE IF (cam=1) THEN P = (1/(1 + EXP(-Y1))) - (1/(1 + EXP(-Y2))); ELSE IF (cam=2) THEN P = (1/(1 + EXP(-Y2))); LL = LOG(P); MODEL cam012 ~ GENERAL(LL); * Random intercept and linear slope; RANDOM U0 U1 ~ NORMAL([0,0],[V11,V21,V22]) SUBJECT=patient_ID; RUN; Ordinal Longitudinal Example 5 of 7

NOTE: GCONV convergence criterion satisfied (and no error messages in the log!) -2 Log Likelihood 932.9 AIC (smaller is better) 954.9 AICC (smaller is better) 955.4 BIC (smaller is better) 982.5 Parameter Estimates Parameter Estimate Error DF t Value Pr > t Alpha Lower Upper Gradient B01 0.8466 0.2425 89 3.49 0.0008 0.05 0.3648 1.3284 1.426E-7 B02-1.7277 0.3085 89-5.60 <.0001 0.05-2.3406-1.1147 1.81E-8 B11day -0.1059 0.06039 89-1.75 0.0829 0.05-0.2259 0.01409-9.9E-7 B21mmse -0.08176 0.03641 89-2.25 0.0272 0.05-0.1541-0.00941-8.25E-6 B31mmseday -0.01454 0.009330 89-1.56 0.1226 0.05-0.03308 0.003996-0.00004 Hoffman Psyc 945 Example 6b Fit statistics from Proportional version of same model: -2 Log Likelihood 941.85 2ΔLL(3) = 9.05 AIC (smaller is better) 957.85 AICC (smaller is better) 958.14 BIC (smaller is better) 977.94 B12day -0.1515 0.07616 89-1.99 0.0498 0.05-0.3028-0.00016 6.501E-7 B22mmse -0.06759 0.03804 89-1.78 0.0790 0.05-0.1432 0.007991 1.199E-6 B32mmseday -0.00254 0.009449 89-0.27 0.7886 0.05-0.02132 0.01623 6.219E-6 V11 1.2730 0.6449 89 1.97 0.0515 0.05-0.00834 2.5543 1.418E-7 V21-0.1614 0.1304 89-1.24 0.2190 0.05-0.4204 0.09765 1.014E-6 V22 0.06281 0.03859 89 1.63 0.1072 0.05-0.01388 0.1395 9.457E-7 2ΔLL(3) = 9.05, p <.05, suggesting that each submodel needs its own set of fixed effects. However, the interaction of MMSE*day is not significant for either submodel, and could be removed. 2d) Ordinal Mixed Non-Proportional Model: Removing MMSE*Day Interactions Logit yti 0 001 101 U1i Dayti 011 MMSEi 16 U0i Logit y 1 U Day MMSE 16 U ti 002 102 1i ti 012 i 0i TITLE1 "Ordinal Mixed Non-Proportional Model for MMSE - No MMSE*Day"; PROC NLMIXED DATA=alldata METHOD=GAUSS TECH=QUANEW GCONV=1e-12; * Must list all parms to be estimated here with start values; * B01 and B02 = intercepts for each equation; * B's = fixed effects; * V's = variance components in order of G matrix; PARMS B01=.6 B02=-1.6 B11day=0 B21mmse=0 B12day=0 B22mmse=0 V11=1 V21=-.2 V22=.05; * Linear predictive model - written as single-level equation; Y1 = B01 + U0 + B11day*day + B21mmse*mmse16 + U1*day; Y2 = B02 + U0 + B12day*day + B22mmse*mmse16 + U1*day; * Model for probability of response - writing it the shorter way; IF (cam=0) THEN P = 1 - (1/(1 + EXP(-Y1))); ELSE IF (cam=1) THEN P = (1/(1 + EXP(-Y1))) - (1/(1 + EXP(-Y2))); ELSE IF (cam=2) THEN P = (1/(1 + EXP(-Y2))); LL = LOG(P); MODEL cam012 ~ GENERAL(LL); * Random intercept and linear slope; RANDOM U0 U1 ~ NORMAL([0,0],[V11,V21,V22]) SUBJECT=patient_ID; RUN; NOTE: GCONV convergence criterion satisfied (and no error messages in the log!) -2 Log Likelihood 936.2 AIC (smaller is better) 954.2 AICC (smaller is better) 954.6 BIC (smaller is better) 976.8 Ordinal Longitudinal Example 6 of 7

Parameter Estimates Parameter Estimate Error DF t Value Pr > t Alpha Lower Upper Gradient B01 0.8092 0.2408 89 3.36 0.0011 0.05 0.3307 1.2878 1.971E-6 B02-1.7720 0.2927 89-6.05 <.0001 0.05-2.3536-1.1903-1.14E-6 B11day -0.08586 0.05515 89-1.56 0.1230 0.05-0.1954 0.02372 0.000014 B21mmse -0.1292 0.02455 89-5.27 <.0001 0.05-0.1780-0.08047-3.02E-6 B12day -0.1314 0.06717 89-1.96 0.0536 0.05-0.2649 0.002058-8.47E-6 B22mmse -0.07113 0.02580 89-2.76 0.0071 0.05-0.1224-0.01988-0.00003 V11 1.3819 0.6610 89 2.09 0.0394 0.05 0.06859 2.6952-1.46E-6 V21-0.1654 0.1295 89-1.28 0.2051 0.05-0.4228 0.09203-6.54E-6 V22 0.05569 0.03683 89 1.51 0.1340 0.05-0.01749 0.1289-0.00001 Hoffman Psyc 945 Example 6b Change in Logit across Days by Sub-Model and MMSE 5.00 Sub-Model 0 vs 12: MMSE = 9 Sub-Model 0 vs 12: MMSE = 23 Sub-Model 01 vs 2: MMSE = 9 Sub-Model 01 vs 2: MMSE = 23 4.00 There is a linear relationship between day and the logits. 3.00 2.00 1.00 Logit (Y) 0.00-1.00-2.00-3.00-4.00-5.00 0 1 2 3 4 5 6 7 8 9 10 11 12 Day Since Hospital Admission Change in Probability across Days by Sub-Model and MMSE Sub-Model 0 vs 12: MMSE = 9 Sub-Model 0 vs 12: MMSE = 23 Sub-Model 01 vs 2: MMSE = 9 Sub-Model 01 vs 2: MMSE = 23 1.00 0.90 There is a NON-linear relationship between day and the probabilities. 0.80 0.70 Probability (Y) 0.60 0.50 0.40 0.30 0.20 0.10 0.00 0 1 2 3 4 5 6 7 8 9 10 11 12 Days Since Hospital Admission Ordinal Longitudinal Example 7 of 7

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